Let's see if I understand your question. You have chicks you say were hatched May 30. That would make them almost a year old. So you likely mean they hatched April 30. Right? So they would be about a week old?
If the chick food you are feeding them is a commercial chick starter, it is nearly identical to an all flock feed. In other words, they are interchangeable. You can feed adult chickens chick feed and you can feed baby chicks all flock feed. Because they are practically the exact same thing.
All chickens can eat eggs.
It's a silly notion that baby chicks would be harmed by eating eggs. What do you think nourishes them in the egg during the three weeks they are developing as an embryo in the shell? The egg yolk is their main source of food in the shell, and they are still absorbing it into their bodies for a few days after they hatch. Of course they can eat eggs.